Why does math "work"?

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How can we explain that abstract mathematical structures are successfully applied in the real world?

7 Answers

  1. The answer, in my opinion, is very simple and does not require any special abstractions.

    Mathematical concepts originated (or rather, were found by humans) one way or another from observations of the surrounding world. The concept of a straight line-as a way to get to the goal by the shortest route. The concept of number – as a result of estimating the number of people in a tribe and the number of animals obtained for food. And so on and so forth.

    In addition to the initial mathematical concepts, logic was also required. It was also gradually developed by people. It allowed us to make correct statements as a consequence of already verified ones.

    The first mathematical concepts plus logic-this allowed us to build new statements that were tested in practice. So the building of “primitive mathematics”was gradually built. And then this process expanded and deepened. But its basis is natural objects and concepts. That's why math “works”. For its source is our World.

    Modern mathematical concepts (such as functors) seem infinitely remote from real life. But you can see how they emerged from the concepts with which mathematics began.

  2. Dear colleagues. What is primary: the logic of teaching or the logic of functioning of the object of research? Throw a rock at me, but I don't understand you. What's the point of going down “in the age of mammoths” and pulling on the old convenient tools for a rapidly changing reality? Thousands of studies have been done every year for decades. There is a selection and classification of steps of the scientific method, in which there is a certain place in analytics (seventh at least in a row and beyond) reserved for math. For more information, see the answer to the anonymous question: “How, in simple terms, does a hypothesis differ from a theory?” on my page.

    The first story. At the meeting of the Dissertation Council, the professor of Mathematics said that he had reviewed all the defended and already approved dissertations in the reading room. And I found mathematical errors in almost all of them?! To which the answer came that dissertations, of course, should be checked, including about mathematics, but it is better before the defense!

    What is it, really? Mathematical errors are not always fatal. And if you go from the object under study, they absorb the stages of understanding the problem, visualizing and structuring the object and subject of research, correctly setting hypotheses about the relationship, correctly performing passive and active experiments, collecting, processing, measuring and systematizing data, and so on. As a result, they are summed up in the assessment of the convergence of the results of mathematical calculation, experiment, and practical use.

    The second story. Along the way, the first-time graduate students present asked the question:”What kind of mathematics should I master in order to conduct industrial scientific research?” After a long pause, the answer came: “arithmetic … mathematical statistics… and then individually according to the situation.” In the context of large-scale development targets, tools for multi-level functional-target mathematical description of systems, corporate FSA, and program-target planning and management are useful. But they are purely individual in relation to the object under study. And they cannot be recommended indiscriminately for all cases of life without a serious, time-consuming justification of the structures of the functional environment and the development of models.

    Experts first check the validity of the logic of reasoning about the composition, structure and processes of the studied area, and then measure and calculate. And mathematics works reliably because it was possible to delve quite accurately into the problem under study and feel it, to visualize and structure it, conceptualize it, and correctly put hypotheses about connections. The more conscientiously this is done, the more adequate, simpler and more reliable the mathematical interpretation of the phenomenon. The more reliable its databases and knowledge bases are used in planning, forecasting and management practices. Mathematics works if it displays the logic of the functioning of the research area, the primary features of the object and subject of research.

    Sincerely, Alexander.

  3. A mathematical scientist works as a bio-robot and applies concepts that are built into it by Nature and that are aimed at solving general problems of survival and development.

    People are part of Nature, so Nature helps people find optimal solutions to problems.

  4. Within the framework of the experimental realism paradigm created by the philosopher Mark Johnson and the cognitive linguist George Lakoff, mathematics can be understood as the science of the pre-conceptual structures of human bodily experience-kinesthetic image schemes. For example, you can give such correspondences between basic mathematical concepts and related figurative schemes (or other basic concepts):

    entity — A REAL OBJECT OR CREATURE

    relation-LINK (LINK)

    continuity, sequence — PATH, TRAJECTORY (MOVEMENT)

    order — DIRECTION

    limited — BORDERS (RECEPTACLES)

    multiplier — or decomposition, division) – PART-WHOLE, DIVISION

    prime number-PART (NOT INCLUDING OTHER PARTS)

    limitedness, finiteness-SEPARATENESS, BEGINNING-MIDDLE-END

    chain-REPEAT, LINK

    equality (quantities) – EQUILIBRIUM

    identity — THE RELATIONSHIP OF TWO ENTITIES TO ONE

    unit (of measurement) – MASS, SEPARATENESS, BALANCE

    cyclic — RING, DIRECTION

    density, compactness-SEPARATION (NO SEPARATION)

    AGENT operator

    operation — CHANGE TO ANOTHER ENTITY

    single element (1 X a = a) — AGENT THAT DOES NOT CAUSE A CHANGE

    zero element (0 X a = 0) — AGENT THAT CAUSES A CHANGE IN THE AGENT ITSELF

    (Taken from George Lakoff's Women, Fire, and Dangerous Things.)

    The answer about the efficiency of mathematics in reality also follows from this: it is not a science about out-of-body “ideal platonic entities”, but just the opposite-mathematical structures arose as a result of understanding the surrounding reality in terms of basic concepts of cognition. Creating this consistent and deep understanding required the careful and hard work of many generations of mathematicians.

  5. Mathematics “works” because its models are built on the basis of properties and relations of the real world. For example, since ancient times, geometry has developed on the basis of the practical need for land surveying and construction. Arithmetic – based on trade, production, accounting, etc. Therefore, mathematical models reflected spatial, temporal, and other forms of the real world. Sometimes mathematics can develop models even before the actual practical needs for doing so (complex numbers) have been formulated. But often later it turned out that these models still reflect some real relations in the world (the laws of alternating current). It is possible that the ability to model spatiotemporal and quantitative forms is inherent in humans at the genetic level.

  6. Yes, humans and others (including unicellular ones) reason (form adequate reactions to life) only with mathematical images, which are objects of complex (to varying degrees) symmetry. (The language is also organized in terms of grammar, rules for organizing symmetric images, and the structure of Mat derivatives. Analysis).

    That is, they arise from a mathematical identity. At the same time, binary relations do not contain binary…, much less complex constructions from object relations.

    That is, at the same time, the fool remains so in Africa itself, organizing its own culture – philosophy. At the same time, it feeds on those who have more adequate mathematical images of consciousness (while doing casuistry, etc.). idiocy, self-satisfying and raping others, including here with their rotten gibberish…).

  7. This is a very interesting question. You can also set it for logic.
    I believe that our logical and mathematical abstractions reflect the properties of the external world. The world is to some extent “logical” and “mathematical” in itself. So mathematics in the part that finds application in the natural sciences is far from an abstraction. But exactly what is “in part”!
    How does the world turn out to be “logical” and “mathematical”?
    Dialectics, both Hegelian and materialist, was for a long time a” decoy ” that gave hope for the solution of this question. In the Science of Logic, Hegel also considers relations related to quantity, trying to deduce mathematics from the general “logic” of the world.
    Unfortunately, the dialectic is either incomplete and / or misunderstood, or it is a false path. It did not give any practical results.
    However, there are other ways to find logic and mathematics in the universe itself.
    Here are two articles that claim that mathematics works in physics because both physics and mathematics reflect such properties of the universe as symmetry (Noether's theorem is related to this in physics): https://m.habr.com/ru/post/390201/https://m.habr.com/ru/post/409045
    /
    Another way to solve the question might be that mathematics “works” only in the part where it follows observations. After all, for example, the bill was not invented in some abstract way, it was taken out of practice. It is not surprising that mathematics and geometry “work” everywhere within the school curriculum. They were actually invented for this purpose.
    Jaglom points out in this pamphlet (Chapter 4) that “pure” mathematics and geometry emerged after mathematics and geometry describing the real world, real bodies. Interesting is his term “geometry-physics”, which he introduces in contrast to” abstract ” geometry.
    https://www.mathedu.ru/text/yaglom_matematika_i_realnyy_mir_1978/p62/
    The more abstract and arbitrary mathematics becomes, the further it is from the natural sciences.
    And, indeed, not all math “works”.
    Here are some examples of abstractions that are not used in physics:
    https://vuzlit.ru/927739/matematika_realnyy
    Here's a more radical take on the usefulness of abstract mathematics: https://proza.ru/2015/11/16/160
    Mathematics is full of contradictions and paradoxes. For example, here: https://m.habr.com/ru/post/167583/
    And how, for example, to understand such a place in probability theory that the probability that a continuous random variable will take a specific value = 0!?
    In Russian, this means that the NSV cannot take on any meaning! However, this consequence is not generally accepted. Although, in fact, this means that mathematicians stupidly do not know how to calculate the probabilities of NSV, except on some interval.
    In addition, mathematics, oddly enough, is not always able to give accurate results. For example, Alexandrov points out that the dimensions of bodies cannot be determined precisely, referring to the Pythagorean theorem (Section 1):
    https://beskomm.livejournal.com/107649.html
    In my opinion, mathematics has failed even with Zeno's aporia: to “resolve” the aporia about Achilles and the tortoise, series convergence is used. But, let me say, the conclusion that the series converge follows from the AXIOM of completeness and the existence of an upper and lower bound. That is, what Zeno demanded to prove is accepted as an axiom!
    Do mathematical abstractions such as infinite divisibility have a place in reality? Unlikely. In physics, there is a Planck length, but it is still impossible to divide further.
    Does actual infinity hold? If the universe is closed, then no.
    I will repeat my assumption again: mathematics and logic work only in the part where they reflect the properties of the real world. And these properties are found out by the natural sciences, for which mathematics is not the queen, but the language. Perhaps in the future, a PHYSICAL “theory of everything” will provide an answer to which mathematics works and which doesn't.
    Lobachevsky called his geometry “imaginary”. It is possible that most of the “imaginary” mathematics will remain so

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